Do we need an alternative to local gauge coupling to electromagnetic fields?

The local gauge coupling through the recipe $\partial_\mu \psi \to \partial_\mu \psi + iqA_\mu \psi$, that works so well with Dirac spinors in QED and in the gauge theories of the Standard Model, has a peculiarity when applied to scalar fields: it generates in the Lagrangian a coupling term $J_\mu A^\mu$ in which $J_\mu$ does not coincide with the conserved N\"other current associated to the global gauge symmetry. This is not an inconsistency, just a feature that appears when working out the locally gauge invariant action, and which ensures that the correct conserved current is the source of the gauge field. What would happen then if we were to assume for the scalar field the same coupling $J_\mu A^\mu$ through a conserved current which holds for spinor QED and classical electrodynamics? The consequence is that one is forced in that case to renounce to the principle of local gauge symmetry and must thus consider the electromagnetic (e.m.) field to be described by electrodynamic theories compatible with that lack of invariance, like the extended electrodynamics by Aharonov-Bohm. No differences with the usual theory appear for fermion systems when strict local charge conservation applies. In particular, if we consider the non-relativistic quantum theory as the low-energy limit of the relativistic theory, we would expect no modifications of Schr\"odinger equation when applied to fermion systems. However, when scalar boson systems are considered, like Cooper pairs quasi-particles in superconductors, in the new formulation the e.m.\ fields include a source, additional to the usual conserved four-current, and, besides, the corresponding Schr\"odinger equation acquires a new term, proportional to $\mathbf{A}^2$, which can lead to observable consequences, like ... (length limit reached, see PDF)